By a (bivariate) mean we understand a function m defined on ((0,infty)thatsatisfies)) thet satisfollowingfollowing double inequality: forall a,b>0, quad min a,b leq m a,b leqmax a,b).
By a solution of (1), (2) we understand a function u ∈ X such that u ′ ∈ X and (1), (2) hold true.
By a solution of problem (2.1), we understand a function u ∈ E with φ p ( u ′ ) absolutely continuous which satisfies problem (2.1).
where g, h : Z → [ 0, ∞ ) and f : Z → R are T-periodic functions, λ, μ > 0. By a solution to (1.1) we understand a function u ∈ E : = { u : Z → R | u ( t ) = u ( t + T ) } satisfying (1.1). Special cases of Eq. (1.1) are (1.2) (1.3) (1.4). In the related literature, it is said that (1.3) has an attractive singularity, whereas (1.4) has a repulsive singularity.
By a solution of (1.1) we understand a function x : [ a, b ] → X such that ∫ a b d [ A ] x exists and (1.1) is true for all t ∈ [ a, b ].
By a solution of (4.13) we understand a function y : [ a, b ] T → R m satisfying the integral equation y ( t ) = y ˜ + ∫ a t [ P ( s ) y ( s ) + h ( s ) ] Δ s, t ∈ [ a, b ] T, where the integral is the Riemann Δ-integral defined e.g. in [22].
By a solution of (1) we understand a function x ∈ C ( [ t 1 − ρ 1, ∞ ), R ), for some t 1 ⩾ t 0, such that x ( t ) − P 1 ( t ) x ( t − τ ) is n − 1 times continuously differentiable, r ( t ) ( x ( t ) − P 1 ( t ) x ( t − τ ) ) ( n − 1 ) is continuously differentiable on [ t 1, ∞ ) and (1) is satisfied for t ⩾ t 1. Similarly, let ρ 2 = max { τ, d }.
By a solution of (2) we understand a function x ∈ C ( [ t 1 − ρ 2, ∞ ), R ), for some t 1 ⩾ t 0, such that x ( t ) − P 1 ( t ) x ( t − τ ) is n − 1 times continuously differentiable, r ( t ) ( x ( t ) − P 1 ( t ) x ( t − τ ) ) ( n − 1 ) is continuously differentiable on [ t 1, ∞ ) and (2) is satisfied for t ⩾ t 1. Finally, let ρ 3 = max { b, d }.
Although the λ-calculus is 'about' calculating with functions by substituting values for arguments, this simple point of view cannot support a semantics for the (untyped) λ-calculus if by 'function' we understand, as is standard in set theory, a relation R such that for every pair (x,y) and (x,z) in R with the same first component x we have y = z.
By a Filippov solution of (4), we understand a function (x cdot colon J rightarrowmathbb{R}) with absolutely continuous derivative, satisfying problem (5), almost everywhere on J.
"These multi-symptomatic disorders caused by a defect in cilium function are really things we understand very poorly," Nachury says in the video above.
